For some families, the \(\mathcal {F}\) -transitivity remains on some well-known notions in the topological dynamic.
For instance, if \(\mathcal {F}\)  is the family of infinite sets, then the \(\mathcal {F}\) -operators are precisely the operators which are topologically transitive.
Suppose \(\mathcal {F}\)  is the family of cofinite sets. In that case, the \(\mathcal {F}\) -operators are the operators who are mixing, see [1]}, [2]}, [3]}.
If \(\mathcal {F}\)  is the family of thick sets (those that contain arbitrarily long intervals), then the \(\mathcal {F}\) -operators are the operators who are weak mixing, see [4]}, [5]}, [6]}. When \(\mathcal {F}\)  is the family of synthetic sets (those that have
bounded gaps) then the \(\mathcal {F}\) -operators are those which are ergodic.
